Summary: 1. Sets, relations and functions.
1.1 Set theory. We assume the reader is familiar with elementary set theory as it is used in mathematics
today. Nonetheless, we shall now give a careful treatment of set theory if only to to allow the reader to
become conversant with our notation. Our treatment will be naive and not axiomatic. For an axiomatic
treatment of set theory we suggest that the reader consult the Appendix to [K] where one will nd a concise
and elegant treatment of this subject as well as other references for this subject.
By an object we shall mean any thing or entity, concrete or abstract, that might be a part of our
discourse. A set is a collection of objects. Whenever A is a set and a is one of the objects in the collection
A we shall write
a 2 A
and say a is a member of A. A set is determined by its members; that is, if A and B are sets then
(1) A = B if and only if for every x, x 2 A , x 2 B;
this is an axiom; in other words, it is an assumption we make.
The most common way of dening sets is as follows. Suppose P (x) is a formula in the variable x. We
will not go into just what this might mean other than to say that (i) if y is a variable then P (y) is a formula
and (ii) if a is an object and if each occurrence of x in P (x) is replaced by a then the result P (a) is a
statement; we will not go into what this means other than to say that statements are either true of false. At
any rate, it is an axiom that there is a set
fx : P (x)g